Two-dimensional improved attribute reductions based on distance granulation and condition entropy in incomplete interval-valued decision systems

Attribute reductions rely on knowledge granulation and information measurement. Aiming at incomplete interval-valued decision systems (IIVDSs), an attribute reduction (with the FSR-AR/HS-AR algorithm) emerges by combining distance granulation and condition entropies; however, its distance measurement has defects for maximum and minimum fillings, while its condition entropy has promotion space on information enrichment. This paper utilizes FSR-AR/HS-AR to make the two-dimensional improvements of distance granulation and condition entropy to improve IIVDS-driven attribute reductions. Concretely, distance measures and similarity degrees are corrected via range completion and statistical enhancement, and the condition entropy is deepened by replacing credibility with coverage-credibility, so 2 x 2 = 4 reduction algorithms extend and improve FSR-AR/HS-AR. First, new maximal and minimal distances are defined via complete search and statistical optimization of missing values, so an amended fuzzy alpha-similarity relation induces knowledge granulation. Then, improved informational, conditional, joint entropies, and mutual information are established via coverage-credibility, and they obtain system equations and granulation nonmonotonicity. Furthermore, the two-dimensional improvements generate 2 x 2 = 4 attribute reductions, and the corresponding heuristic algorithms include FSR-AR/HS-AR and three improved algorithms. Finally, improvements in uncertainty measures and reduction algorithms are validated via table examples and data experiments, and the three novel algorithms outperform FSR-AR/HS-AR and contrastive algorithms for classification performances.